Convergence of Linear Approximation of Archimedean Generator from Williamson's Transform in Examples
Abstract
We discuss a new construction method for obtaining additive generators of Archimedean
copulas proposed by McNeil and Neslehova [8], the so-called Williamson n-transform,
and illustrate it by several examples. We show that due to the equivalence of convergences
of positive distance functions, additive generators and copulas, we may approximate
any n-dimensional Archimedean copula by an Archimedean copula generated by
a transformation of a weighted sum of Dirac functions concentrated in certain suitable
points. Specically, in two-dimensional case this means that any Archimedean copula
can be approximated by a piece-wise linear Archimedean copula, moreover the approximation
of the generator by linear splines circumvents the problem with the non-existence
of explicit inverse.
copulas proposed by McNeil and Neslehova [8], the so-called Williamson n-transform,
and illustrate it by several examples. We show that due to the equivalence of convergences
of positive distance functions, additive generators and copulas, we may approximate
any n-dimensional Archimedean copula by an Archimedean copula generated by
a transformation of a weighted sum of Dirac functions concentrated in certain suitable
points. Specically, in two-dimensional case this means that any Archimedean copula
can be approximated by a piece-wise linear Archimedean copula, moreover the approximation
of the generator by linear splines circumvents the problem with the non-existence
of explicit inverse.